J. Phys. Chem. 1996, 100, 15555-15561
15555
A Simple Model for Ac Impedance Spectra in Bipolar Membranes A. Alcaraz and P. Ramı´rez Departamento de Ciencias Experimentales, UniVersidad Jaume I de Castello´ n, Apdo 224, E-12080 Castello´ n, Spain
S. Mafe´ * Departamento de Termodina´ mica, Facultad de Fı´sica, UniVersidad de Valencia, E-46100 Burjassot, Spain
H. Holdik Institut fu¨ r Prozesstechnik, UniVersita¨ t Saarbru¨ cken, D-66123 Saarbru¨ cken, Germany ReceiVed: April 24, 1996; In Final Form: June 25, 1996X
A new model accounting for the ac impedance spectra of synthetic ion-exchange bipolar membranes is presented. The theoretical approach is based on the Nernst-Planck and Poisson equations and applies some of the concepts used in the semiconductor pn junctions to the case of a bipolar membrane. The results presented are the current-voltage curves and the impedance spectra at electric currents above the limiting current. It is shown that the model is able to identify the main contributions to the bipolar membrane impedance and gives valuable information about the bipolar junction structure and its influence on the characteristic parameters involved in the field-enhanced water dissociation phenomenon.
Introduction A bipolar membrane (BM) contains an anion-exchange layer and a cation-exchange layer joined together in series. Much of the interest of this system concerns the possibility of employing bipolar models to explain both the behavior of biological membranes showing current rectification1,2 and negative/positive charge distributions3,4 and the performance of synthetic BM’s to produce acids and bases from their corresponding salts by electrodialysis.5-8 The ability of the BM to generate an excess of H+ and OH- ions when the membrane is reverse biased is the most important feature in this case. In both cases, a deep knowledge of the properties of ion transport and water dissociation in BM’s is required. The characterization of BM’s has been carried out by studying the current-voltage (I-V) curves under forward9-11 and reverse5,9-14 bias conditions and the membrane potential.15,16 Such studies provide valuable information, but still a deeper knowledge of the characteristic parameters involved in the phenomenon of water dissociation is needed. In this sense, ac impedance techniques could give more insights into membrane parameters recognized to be crucial in the BM performance,5,17,18 such as the electric field at the membrane interface and the capacitance of the bipolar junction where the water dissociation is taking place. It is generally agreed that one of the problems of current BM technology (in addition to that of membrane stability) is the reduced information available on the crucial phenomenon occurring in the membrane: the electric field enhanced water dissociation. This phenomenon could also be operative in biological membranes.4 Ac impedance techniques are currently used in the study of a wide variety of electrochemical phenomena.19 Complex impedance plots are of particular importance for charged membranes, and several theoretical models are available.20-23 The problem of the ac impedance in BM’s has also been treated * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, September 1, 1996.
S0022-3654(96)01187-2 CCC: $12.00
extensively in the literature. Chilcott et al.24,25 have presented recently a theoretical model accounting for the impedance spectra of BM’s based on previous analyses.26-28 The model, which considers only the salt ions and cannot thus be applied to the case of water dissociation, seems to be useful for the description of biological systems29 over a wide range of frequencies. Other attempts30,31 have been focused on the study of the ac spectra of synthetic BM’s separating acid-base solutions by using a model characteristic of the electrode/ solution interface. Though the agreement between theory and experiment appears to be good, the results do not provide a clear relationship between the characteristic parameters of the reaction responsible for the electric field enhanced water dissociation and the electric field in the membrane junction. In this paper, we present a new, simple model for the ac impedance spectra in BM’s under current flow in reverse polarization which takes into account explicitly the water dissociation phenomenon. The theoretical approach is based on the Nernst-Planck and Poisson equations and reproduces the results obtained by previous models28,30 taking the appropriate limiting cases. The crucial part of the paper is the setting up of the model for the field dependence of the rate of dissociation of water in the space charge region existing at the interface between the two ion-exchange layers of the bipolar membrane. The model assumes that the electric field that appears in this interface when the membrane is reverse biased enhances the generation rate of H+ and OH- ions, while the recombination rate remains basically unaffected. Also, we show that the model can explain some of the experimental trends observed in impedance measurements with BM’s.32 It is expected that proposing a model simple enough to identify the main contributions to the BM impedance, but yet capable of describing the water dissociation in the BM junction, could contribute to a better characterization of BM’s and, ultimately, to a deeper understanding of the field-enhanced water dissociation phenomenon. © 1996 American Chemical Society
15556 J. Phys. Chem., Vol. 100, No. 38, 1996
Figure 1. Sketch of the system under study. Equilibrium concentration profiles (a) and electric potential profiles under forward (b) and reverse (c) polarization. The space charge region lies from x ) -λN to x ) λP. The thickness of this region has been exaggerated.
Theoretical Model The system considered is shown schematically in Figure 1a. The BM separates two solutions of the same uni-univalent electrolyte. x is the coordinate along an axis normal to the membrane surfaces. Subscripts K ) L, R, N, and P refer to the left and right bulk solutions and to the anion and cation exchange layers of the BM, respectively. ciK stands for the concentration of the ith species in the region K (i ) 1 for salt cations, i ) 2 for salt anions, i ) 3 for hydrogen ions, and i ) 4 for hydroxyl ions). The fixed charge concentrations XK (K ) N, P) are assumed to be constant throughout each layer. Since in synthetic BM’s the thicknesses dK (K ) L, R) of the ionexchange layers are much greater than the typical Debye length, the electroneutrality condition holds in every bulk phase.22 The whole system is assumed to be isothermal and free of convective movements. The bathing solutions are supposed to be perfectly stirred, and therefore the effects of diffusion boundary layers adjacent to the membrane/solution interfaces are neglected. When two layers with fixed charges of opposite sign are brought in close proximity, the space charge regions in the solution phase at the boundary of each layer interact, and as the distance of separation is reduced, the space charge regions are forced further in each layer. In the limit, when the distance of separation is zero, the space charge resides completely in the opposite fixed charge layer.2 This gives place to a space charge region almost completely devoid of mobile ions, which in solid state physics is known as an abrupt junction (extending from x ) -λN to x ) λP in Figure 1a). The dc current-voltage (I-V) characteristics of BM’s have been described extensively in the literature.9,10,14 They have certain analogies with semiconductor pn junctions: under forward polarization (with the cathode facing the anion exchange layer, Figure 1b), the cations first enter the cation exchange layer and the anions the anion exchange layer. These salt ions accumulate from both sides in the median plane of the bipolar membrane and the concentrated salts swamps the Donnan exclusion. Therefore, the ions are able to cross into the second,
Alcaraz et al. previously excluding layer of the sandwich and leave the membrane on the other side.9 The BM shows a very low resistance and a rapid increase of the current with the applied voltage. Under reverse polarization (with the cathode faced to the cation exchange layer, Figure 1c), the ions from the solution first have to enter the layers which exclude them, against repulsion, while the counterions of these layers are being pulled out by the electric current. This leads to an increase in the space charge region thickness and in the electric field in this region. The experimental trends observed are first a region of high resistance, characterized by a limiting current, and then a second region showing a rapid increase of the current with the applied voltage. A sketch of the electric potential profiles in the bipolar membrane is shown in Figures 1b,c: forward bias (Figure 1b) and reverse bias (Figure 1c). Measurements of the pH of the solutions close to the membrane show that in this second region most of the current is carried by the H+ and OH- ions generated probably at the membrane junction. It is believed that the mechanisms responsible for the generation of the H+ and OHions are protonation and deprotonation reactions between the ionizable groups fixed to the membrane matrix and water.12,13,33-35 The fact that this phenomenon, known as “water splitting” or “electric field enhanced (EFE)” water dissociation, takes place only at high enough applied voltages under reverse polarization suggests that the high electric field which exists in the space charge region at the junction of the BM under these conditions could play an important role in the acceleration of water dissociation.13,14,34-36 It has also been recognized that the presence of traces of several heavy metal hydroxides can increase the performance of the ion-exchange membranes in the generation of the H+ and OH- ions.32,37,38 The general trends above mentioned can be explained on the basis of a theoretical model accounting for ion transport and EFE water dissociation in BM’s previously developed.10,14 In particular, under reverse bias, the I-V curve of the BM in steady state conditions can be written in the form
I ) (ILS + ILW)[exp(FV/RT) - 1] - Id
(1)
where I is the current density passing through the system, V is the applied voltage (V < 0 under reverse bias), and F, R, and T have their usual meanings. The constants ILS and ILW are defined in the form
ILS ≡ F[D2Nc2N(-dL)/dL + D1Pc1P(dR)/dR]
(2)
ILW ≡ F[D3Pc3P(dR)xχP/D3P coth(dRxχP/D3P) +
D4Nc4N(-dL)xχN/D4N coth(dLxχN/D4N)] (3)
and can be interpreted, respectively, as the limiting current densities carried by the salt ions and by the H+ and OH- ions generated under normal conditions (i.e., when no external electric field is applied under reverse polarization). In eq 1, Id is the current density due to the EFE water dissociation. In eqs 2 and 3, DiK stands for the diffusion coefficient of the ith species in region K, ciK(x), x ) -dL,dR, are the concentrations of the ith species at the interface solution/Kth layer, and
χP ≡ kr0c4P(dR)
(4a)
χN ≡ kr0c3N(-dL)
(4b)
where kr0 is the rate constant for the recombination of water ions when no electric field is applied. The concentrations of the ionic species at the membrane-solution interfaces that appear in eqs 2-4 can be calculated in terms of the external bathing
Ac Impedance Spectra in Bipolar Membranes
J. Phys. Chem., Vol. 100, No. 38, 1996 15557
concentrations by using the Donnan equilibrium equation and the electroneutrality condition in each ion-exchange layer, yielding
u(x;t) ) u(x) + uˆ (x)ejωt
ciL ciN(-dL) ) [ (X /2)2 + (c1L + c3L)2 + c1L + c3L x N XN/2], i ) 1, ..., 4 (5a)
i+1
(-1) ciP(dR) )
ciR [ (X /2)2 + (c1R + c3R)2 + c1R + c3R x P (-1)iXP/2], i ) 1, ..., 4 (5b)
The term Id of eq 1 prevails at high reverse applied voltages. According to refs 13, 14, and 35, this term can be written in the form
Id ≈ Fkd(E)nλ ) Fk0d exp(RFE/RT)nλ
E)
(
(
)
XP + XN 2r0 (-V) F XPXN
)
XPXN 2F (-V) r0 XP + XN
1/2
(7)
1/2
(8)
are respectively the total thickness of the space charge region and the maximum value of the electric field in the same region. In eqs 7 and 8, 0 is the electric permittivity of free space and r is the dielectric constant of the bipolar junction. Equation 6 can be seen as a kind of “absolute rate theory” with the field helping the generation rate in the forward direction.22,39 Since the space charge region is very thin and the H+ and OH- ions are swept away in opposite directions under reverse bias, the recombination term that should appear in eq 6 can be neglected to a good first approximation.39 Substitution of eqs 2-8 into eq 1 allows for the calculation of the dc I-V curve of a bipolar membrane under reverse polarization in terms of the concentration of the bathing solutions and the characteristic parameters of the membrane. If V ) 0, both the first term in eq 1 and Id vanish (see eqs 6-8). When the applied voltage under reverse polarization is small, the term Id is also small, and the current attains the limiting value (ILS + ILW). If the applied voltage becomes greater, the term Id increases rapidly and most of the current is then carried by the H+ and OH- ions generated at the median membrane interface. Experimental data obtained for several BM’s under different conditions have been explained using eq 1.10,14 With some modifications, the model can also be used to describe the I-V characteristics of the BM under forward polarization10,40 as well as the dependence of the membrane potential on the concentration.16 We shall consider now small perturbations of the steady state model described above in order to derive the ac impedance characteristics of the bipolar junction. When a sinusoidal perturbation of angular frequency ω is superimposed on the dc signal, a steady state variable u(x) of the system (which could denote a concentration, an electric potential, an ion flux or an electric field) depends on both the position x and the time t. If
(9)
where uˆ (x) is the amplitude of the ac response and j ≡ x-1. We shall assume that the dc component u(x) still verifies the steady state equations. The basic equations describing the transport of ions through the BM are the one-dimensional Nernst-Planck equations
JiK ) -DiK
[
]
∂ciK ∂φ + (-1)i+1ciK F/RT , i ) 1, ..., 4; K ) ∂x ∂x N, P (10)
the electroneutrality condition in each ion-exchange layer
c1K + c3K ) c2K + c4K ( XK, K ) N, P
(6)
where kd(E) is the forward rate constant of the net reaction responsible for the EFE water dissociation, k0d is the forward rate constant of the reaction when no external electric field is applied, n is the concentration of the active sites where the reaction is taking place, R is a characteristic parameter having the dimensions of length that can be interpreted as the effective reaction distance for dissociation (typically R ∼ 10-10 m35,39) and
λ ≡ λN + λP )
the amplitude of the perturbation is small, the perturbed variable u(x;t) can be written using complex notation in the form
(11)
and the continuity equations
∂JiK ∂ciK + ) 0, i ) 1, 2; K ) N, P ∂x ∂t ∂JiK ∂ciK + ) kdn - krc3Kc4K, i ) 3, 4; K ) N, P ∂x ∂t
(12a) (12b)
Here JiK is the flux of species i in the K region, φ is the electric potential, and kr is the rate constant for the recombination of the H+ and OH- ions. In eq 11, the plus sign applies for K ) N, and the minus sign for K ) P. We shall consider separately the transport problem in each of the bulk exchange layers of Figure 1. Anion Exchange Layer. In the region λP < x < dR, the minority carriers are the salt cations and the hydrogen ions. For these ions, the concentration gradient term is much greater than the potential gradient term, and eqs 10 can be written in the form
∂ciP JiP ≈ -DiP , i ) 1, 3 ∂x
(13)
Since very high electric fields causing EFE water dissociation are expected to occur only in the space charge region, eq 12b gives approximately
∂JiP ∂ciP + ≈ k0dn - kr0c3Pc4P, i ) 3, 4 ∂x ∂t
(14)
Substitution of eq 9 with u(x;t) ) JiP(x;t) and u(x;t) ) ciP(x;t) in eqs 12a, 13, and 14, and further separation of the alternating components gives the following differential equations for the perturbation term of the minority carrier concentrations
d2cˆ iP 2
dx
-
1 cˆ iP ) 0, i ) 1, 3 LiP2
(15)
where
L1P2 ≡ D1P/jω
(16a)
L3P2 ≡ D3P/(jω + χP)
(16b)
Assuming that the perturbation term of the minority carrier
15558 J. Phys. Chem., Vol. 100, No. 38, 1996
Alcaraz et al.
concentration vanishes far from the junction, the boundary conditions that should be used to solve eqs 15 are
capacitor. Thus, the discontinuity in Jˆ 2 can be written in the form24,28
cˆ iP(dR) ) 0, i ) 1, 3
FA[Jˆ 2N(-λN) - Jˆ 2P(λP)] ) jωCVˆ
(17)
which yield
(27)
where C is the geometrical capacitance of the junction1
sinh[(dR - x)/LiP] cˆ iP(x) ) cˆ iP(λP) , i ) 1, 3 sinh[(dR - λP)/LiP]
The term cˆ iP(λP) can be determined as follows. The steady state boundary conditions used for minority carriers in both pn junctions41 and BM’s2,42 are
ciP(λP) ) ciP(dR) exp(FV/RT), i ) 1, 3
C ) r0A/λ
(18)
(19)
(28)
and A is the membrane area. The discontinuity in Jˆ 4 can be calculated from eq 12b. In the space charge region, the electric field reaches very high values and promotes EFE water dissociation. Therefore, the generation term in eq 12b is much greater than the recombination one, and the continuity equation for i ) 4 reduces to
Taking u(x,t) ) ciP(x;t) and u(x;t) ) V(x;t) in eq 9, and bearing in mind eq 17, eq 19 gives
∂J4 ∂c4 + ≈ kd(E)n ∂x ∂t
ciP(λP) + cˆ iP(λP) ejωt ) ciP(dR) exp[F(V + Vˆ ejωt)/RT], i ) 1, 3 (20)
Substituting u(x;t) ) J4(x;t), u(x;t) ) c4(x;t), and u(x;t) ) E(x;t) in eq 9 into eq 29 and expanding kd(E), we obtain
Since the amplitude of the perturbation is very small compared with the dc signal, eq 20 gives approximately
dJ4 dJˆ 4 jωt + e + jωcˆ 4ejωt ≈ kd(E + Eˆ ejωt)n ≈ dx dx ∂kd jωt ne (30) ∂E
FVˆ FVˆ exp(FV/RT) ) ciP(λP) , i ) 1, 3 (21) RT RT
kd(E)n + Eˆ
cˆ iP(λP) ≈ ciP(dR)
where only the oscillatory terms have been kept. Cation Exchange Layers. A similar treatment to that developed for eqs 13-21 can also be applied for i ) 2 and 4 in the region -dL < x < -λN, which leads to
After dropping the steady terms and deriving kd with respect to E using eq 6, eq 30 yields
RFk0dn dJˆ 4 + jωcˆ 4 ≈ Eˆ exp(RFE/RT) dx RT
sinh[(dL + x)/LiN]
, i ) 2, 4 (22) cˆ iN(x) ) cˆ iN(-λN) sinh[(dL - λN)/LiN] where
(29)
(31)
Integrating eq 31 from x ) -λN to x ) λP, the discontinuity in Jˆ 4 gives
FA[Jˆ 4N(-λN) - Jˆ 4P(λP)] ≈
L2N2 ≡ D2N/jω
(23a)
L4N2 ≡ D4N/(jω + χN)
(23b)
RAVˆ FId RAF2k0dn Vˆ exp(RFE/RT) ) (32) RT RTλ where we assume that cˆ 4 ≈ 0 inside the space charge region and
and
FVˆ FVˆ exp(FV/RT) ) ciN(-λN) , i ) RT RT 2, 4 (24)
cˆ iN(-λN) ≈ ciN(-dL)
The ac electric current density passing through the system is
ˆI ) F[Jˆ 1P - Jˆ 2P + Jˆ 3P - Jˆ 4P]x)λP
Eˆ ≈ -Vˆ /λ
has been calculated by substituting u(x;t) ) E(x;t) and u(x;t) ) V(x;t) in eq 9 into eq 8. The admittance of the junction can now be computed as
(25)
which can be written in the form
ˆI ) F{[Jˆ 1P(λP) - Jˆ 2N(-λN)] + [Jˆ 2N(-λN) - Jˆ 2P(λP)] + [Jˆ 3P(λP) - Jˆ 4N(-λN)] + [Jˆ 4N(-λN) - Jˆ 4P(λP)]} (26) The fluxes of minority ions in eq 26 (first and third term on the right) can be calculated directly by deriving the concentration profiles given by eqs 18 and 22. The second and fourth terms represent the discontinuities of the ac fluxes of ions 2 and 4 through the membrane junction. The discontinuity in Jˆ 2 arises from the fact that the thickness λ of the space charge region (eq 7) changes with time when the applied dc voltage is perturbed by the small ac signal and, therefore, charge is added to or subtracted from the depletion region edges in response to Vˆ . This phenomenon is similar to the charge fluctuations occurring on the metal plates of a
(33)
Y ) A(Iˆ/Vˆ )
(34)
Here we shall concentrate on the analysis of the impedance of the BM for the symmetrical case. This corresponds to dL ) dR ≡ d, XN ) XP ≡ X, DiK ≡ DS (i ) 1, 2; K ) N, P), DiK ≡ DW (i ) 3, 4; K ) N, P), ciK ≡ cS (i ) 1, 2; K ) L, R), ciK ≡ cW (i ) 3, 4; K ) L, R), χN ) χP ≡ χ, L1P ) L2N ≡ LS, L3P ) L4N ≡ LW, and λN ) λP ≡ λ/2. Furthermore, we will focus on synthetic BM’s having thicknesses d . λ/2. In this case, eq 34 gives
Y)
FAd ω/2DSILS(1 + j) exp(FV/RT) coth[dxω/2DS(1 + RT x FA j)] + tanh[dxχ/DW]ILWa exp(FV/RT) × x2RT r0A RFAId coth[dxχ/2DWa] + jω + (35) λ RTλ
Ac Impedance Spectra in Bipolar Membranes
J. Phys. Chem., Vol. 100, No. 38, 1996 15559
Figure 2. Typical current-voltage curve of a bipolar membrane under reverse bias. The limiting current is ca. 4 A/m2, and the EFE water dissociation effects begin around -1 V, approximately.
Figure 3. Curves of -Im(Z) vs log(ν) for different currents in reverse polarization: 20 (s), 40 (- - -), 60 (- - -), 80 (-‚-‚-) and 100 (-‚‚-) A/m2.
where
form
a ≡ {[1 + (ω/χ)2]1/2 + 1}1/2 + j{[1 + (ω/χ)2]1/2 - 1}1/2
(36)
The admittance of eq 35 is composed of four terms. The first and third ones account for the admittance of the BM junction as described in ref 28, where the transport of only the two salt ions was considered. The second term is the so-called Gerischer admittance30,31,43 and corresponds to the contribution of the H+ and OH- ions generated under normal conditions. The last term accounts for the contribution of the ions generated by the EFE water dissociation and is expected to prevail in reverse polarization when the electric current due to these ions is much greater than the limiting current. Since the first and the second terms of the total admittance have an exp(FV/RT) factor, both of them are expected to vanish at high applied voltages under reverse polarization. The validity of the model can be checked by calculating the low frequency limit of the admittance. In this limit, the admittance of the BM junction must be equal to its dc conductance G0, which can be calculated directly from the I-V curve given by eq 1, yielding
dId dI FA(ILS + ILW) exp(FV/RT) ) G0 ) A ) dV RT dV RFAId FA(ILS + ILW) exp(FV/RT) + (37) RT RTλ Since coth(z) ≈ 1/z and a ≈ x2 when ω f 0, it can be easily proven from eqs 35 and 36 that in this limit Y f G0. Results Since most of the available experimental data concerning frequency analysis are usually reported in terms of the impedance (Z ≡ 1/Y), we shall present here some theoretical results of the model for Z. Figure 2 shows the dc I-V curve of a BM under reverse polarization. Typical values14 for the parameters of the model have been taken: d ) 50 µm, X ) 1 M, DS ) 10-10 m2/s, DW ) 10-9 m2/s, cS ) 0.1 M, cW ) 10-7 M, T ) 298 K, r ) 20, R ) 5 × 10-10 m, k0d n ) 7 × 10-3 mol/(m3 s), and kr0 ) 1.11 × 108 m3/(mol s). With these parameters, the limiting current attained by the system is ca. 4 A/m2, and noticeable EFE water dissociation effects occur at voltages above approximately 1 V in reverse polarization. Figures 3-5 show the impedance spectra of the BM at electric current densities in reverse polarization above the limiting current (I ) 20, 40, 60, and 100 A/m2). For such currents, the last two terms of eq 35 dominate, and the real and imaginary parts of the total impedance can be written, respectively, in the
Re(Z) )
G G2 + (Cω)2
(38a)
Im(Z) )
-Cω G + (Cω)2
(38b)
2
where
G≡
RFAId RTλ
(39)
The equivalent circuit for Z consists of a conductance G and a capacitance C connected in parallel. Figure 3 shows a logarithmic plot of Im(Z) vs frequency ν ) ω/2π at the above-mentioned current densities. Both the high- and the low-frequency limits are zero, as observed experimentally.32 The most interesting feature is the maximum attained by the different curves. The position of the maxima can be readily calculated from eq 38b, yielding
νmax )
G 1 RF I ) 2πC 2π RTr0 d
(40)
while the maximum value of -Im(Z) results in
-Im(Z)max )
1 2G
(41)
Equation 40 shows a linear dependence of νmax with the current Id. Since Id is easily measurable (see Figure 2), the slope of eq 40 (taking Id as the abscissa) could give direct information about the quotient R/r. Equation 41 shows that -Im(Z)max decreases with the electric current, as should be expected from the proportionality of G with Id (see eq 39) and the shape of the I-V curve in Figure 2. Combining eqs 40 and 41, the capacitance of the junction can be readily deduced. However, it must be recognized that the assumption of the existence of an abrupt junction in the BM interface at x ) 0 is crucial in this analysis. In this sense, deviations from abruptness of the junction,40 as well as the possible existence of a neutral water layer11,17,18 between the two ion-exchange layers could strongly influence key parameters of the model such as the electric field E and the capacitance C. This could be the reason for the different positions of the maxima usually found in the literature.13,32 In any case, we have shown that the plots in Figure 3 contain relevant information about the junction structure and the EFE water dissociation parameters of the BM, which shows that impedance measurements constitute a suitable electrochemical tool to characterize BM’s. Figure 4 shows a logarithmic plot of Re(Z) vs ν at the same currents as in Figure 3. The curves present zero high-frequency
15560 J. Phys. Chem., Vol. 100, No. 38, 1996
Alcaraz et al.
Figure 4. Curves of Re(Z) vs log(ν) for the same currents as in Figure 3.
Figure 5. Curves of -Im(Z) vs Re(Z) for the same currents as in Figure 3.
limits (the nonzero experimental limits that can be found in ref 32 must probably be adscribed to bulk resistance effects of the experimental setup), a low-frequency limit Re(Z)ωf0, and an inflection point νinf that can be readily determined from eq 38a as
Re(Z)ωf0 ) νinf )
1 ) -2 Im(Z)max G
G ) νmax/x3 2πx3C
(42) (43)
respectively. These curves give similar information as those of Figure 3. Finally, -Im(Z) vs Re(Z) has been represented in Figure 5 for the same currents as in Figures 3 and 4. Each curve is a semicircle of radius 1/2G centered at Re(Z) ) 1/2G (as expected from the equivalent circuit of the bipolar junction). Experimental plots13,32 show a displacement of the center of the curves due probably to bulk resistance effects of the experimental setup. Discussion The problem of the ac impedance spectra in synthetic BM’s under reverse polarization has been analyzed in terms of a simple analytical model based on the theory of charge carriers in solid state pn junctions and on a model accounting for the dc I-V characteristics previously developed. The model gives the lowand high-frequency limits for the admittance that should be expected from the I-V curve in Figure 2. It has also been shown that it is possible to deduce important information about the BM characteristics from the impedance diagrams shown in Figures 3-5. In particular, the position of the maxima in Figure 3 could be used to determine some of the characteristics of the EFE water dissociation and the BM junction (such as the parameter R and the dielectric constant r) that are not directly
deducible from the I-V curves or from membrane potential measurements. It has also been pointed out that the position of the maxima in Figure 3 is closely related to the structure of the bipolar junction. In particular, if the capacitance of the junction decreases (as a consequence for instance of the existence of a thin water layer between the ion-exchange layers), the position of the maxima should move to higher frequencies in Figure 3. This might be the case of the measurements given in ref 32, where the ion-exchange layers which form the BM were simply clamped together. There are some limitations of the theory that should be pointed out. The most important one concerns the applicability of the present model to the limit of very high frequencies. In this case, a significant part of the total BM admittance should be attributable to the regions outside of the space charge region.24,25,27,28 Furthermore, it is not clear to what extent the pn junction theory for admittance can be applied to a BM, where the times of response of the charge carriers should be much higher. In addition, little is known about the possible influence of the high-frequency signal on key parameters of the model, such as the dielectric constant of the junction and the length parameter for the reaction, R. Finally, in order to compare quantitatively experimental measurements to the present theory, additional effects such as the effective (as opposed to geometrical) area of the membrane28 and the existence of diffusion boundary layers at the membrane/solution interfaces should be taken into account. These effects can now be incorporated in each particular experimental situation on the basis of the proposed theory. Acknowledgment. Financial support from the Generalitat Valenciana under Project No. 3242/95 and the DGICYT, Ministry of Education and Science of Spain, under Project No. PB95-0018 is gratefully acknowledged. Glossary A a C ciK ciK(x) cˆ iK(x) DiK dK DS DW E F G G0 I ˆI Id ILS ILW
membrane area (m2) auxiliary dimensionless function geometrical capacitance of the junction (F) concentration of species i (i ) 1, 2, 3, 4) in solution K (K ) L, R) (mol/m3) concentration of species i (i ) 1, 2, 3, 4) in region K (K ) N, P) at position x (mol/m3) ac amplitude of the concentration of species i (i ) 1, 2, 3, 4) in region K (K ) N, P) at position x (mol/m3) diffusion coefficient of species i (i ) 1, 2, 3, 4) in region K (K ) N, P) (m2/s) thickness of layer K (K ) L, R) (m) diffusion coefficient of salt ions (m2/s) diffusion coefficient of species of H+ and OH- ions (m2/ s) electric field in the junction (V/m) Faraday constant (C/mol) conductance of the junction (S) dc conductance of the junction (S) total electric current density passing through the membrane (A/m2) amplitude of the ac electric current density (A/m2) electric current density carried by the H+ and OH- ions (A/m2) limiting electric current density carried by the salt ions (A/ m2) limiting electric current density carried by the H+ and OHions (A/m2)
Ac Impedance Spectra in Bipolar Membranes j JiK Jˆ iK kd k0d kr k0r LiK n R T t u(x) u(x;t) uˆ (x) V Vˆ x XK Y Z
imaginary unit flux of species i (i ) 1, 2, 3, 4) in region K (K ) N, P) (mol/m2 s) ac amplitude of the flux of species i (i ) 1, 2, 3, 4) in region K (K ) N, P) (mol/(m2 s)) forward rate constant of the reactions (s-1) forward rate constant of the reaction when no external electric field is applied (s-1) rate constant for the recombination of the H+ and OHions (m3/(mol s)) rate constant for the recombination of the H+ and OHions when no electric field is applied (m3/(mol s)) characteristic length parameter of species i (i ) 1, 2, 3, 4) in region K (K ) N, P) (m) concentration of active sites where the reaction is taking place (mol/m3) universal gas constant (J/(mol K)) absolute temperature (K) time (s) generic steady state variable of the system at position x perturbed variable at position x and time t amplitude of the ac perturbation at position x total applied voltage (V) amplitude of the ac applied voltage (V) spatial coordinate (m) fixed charge concentration of layer K (K ) N, P) (mol/ m 3) total admitance of the junction (S) total impedance of the junction (Ω)
Greek Symbols R characteristic length of the reaction (m) auxiliary parameter in region K (K ) N, P) (s-1) χK electric permittivity of free space (F/m) 0 dielectric constant r φ local electric potential (V) λ total thickness of the junction (m) thickness of layer K (K ) N, P) (m) λK ν frequency of the applied electric ac current (Hz) position of the inflection point in Re(Z) vs ν (Hz)) νinf position of maximum in -Im(Z) vs ν (Hz) νmax ω angular frequency of the ac current (rad/s)
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